What's the shape of this "addsTo" function ...? Note that in this combinatronics question,
How many lists of 100 numbers (1 to 10 only) add to 700?
I was asking:
For an array of 100 numbers, each 1 to 10 inclusive, and the total is T - how many such arrays exist?
In fact due to the ASTOUNDING answers there, we now know
for T=700 the answer is 1.2 e92
Being curious I now wonder about the shape of this result for various T.
Sadly, all I know are the points T<100, T=100, T=700 (thanks, MSE!), and T=1000.
Many questions arise, is there a T that makes a maximum, is it the same all over except the ends, is it bumpy or erratic ... does it make any difference if T is prime, even, etc ... and, since 19^92 is pretty small and there's only 600 Ts, in fact, is T=700 just freakishly small for some reason (what reason?) ...... or?
 A: According to the Central Limit Theorem, when you add a large number of independent, identically distributed random variables (each variable can be a sum of other random variables), the distribution of the sum tends to a normal distribution whose mean is the sum of the means of the individual distributions, and whose variance is the sum of the variances of the individual distributions.
For example, if we sum $1$ number ($1$-$10$), the distribution looks like
$\hspace{3cm}$
If we sum $2$ numbers ($1$-$10$), the distribution looks like
$\hspace{3cm}$
If we sum $3$ numbers ($1$-$10$), the distribution starts to looks like a normal distribution
$\hspace{3cm}$
If we sum $10$ numbers ($1$-$10$), the distribution looks even more like a normal distribution
$\hspace{3cm}$
Now lets look at the sum of $100$ numbers ($1$-$10$)
$\hspace{3cm}$
The number of ways for $100$ numbers ($1$-$10$) to sum to $700$, which was computed in this answer, is pointed to by the arrow. Note that the range of the sum is $9n$ where $n$ is how many numbers are added; however, the standard deviation is $\sqrt{8.25n}$. Thus, the relative spread gets smaller; that is, the distribution becomes narrower about the mean as $n$ get bigger.
The maximum number of ways to sum to $T$ is at the mean of the distribution; that is, at $T=550$, where the number of ways is
$$
13868117806391314648666325510838589167047653141664\\4888545033078503482282975641730091720919340564340
$$
which is approximately $1.3868117806391314649\times10^{98}$.
